The ratio of the frequencies of the long wavelength limits of the balmer and Lyman series of hydrogen is

Options

A. `27:5`
B. `5:27`
C. `4:1`
D. `1:4`

Explanation

To find the ratio of the frequencies of the long wavelength limits of the Balmer and Lyman series of hydrogen, we can follow these steps:

### Step 1: Understand the Series
- The Lyman series involves transitions where the electron falls to the first energy level (n=1) from higher levels (n=2, 3, 4, ...).
- The Balmer series involves transitions where the electron falls to the second energy level (n=2) from higher levels (n=3, 4, ...).

### Step 2: Identify Long Wavelength Limits
- The long wavelength limit corresponds to the transition with the smallest energy difference, which occurs when the electron transitions from the first level to the second level in the Lyman series (n1=1, n2=2).
- For the Balmer series, the long wavelength limit occurs when the electron transitions from the third level to the second level (n1=2, n2=3).

### Step 3: Use the Rydberg Formula
The Rydberg formula for the wavelength (λ) of the emitted light is given by:
\[ \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
where \( R \) is the Rydberg constant.

### Step 4: Calculate Wavelengths for Lyman and Balmer Series
- For the Lyman series (n1=1, n2=2):
\[ \frac{1}{\lambda_L} = R \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = R \left( 1 - \frac{1}{4} \right) = R \left( \frac{3}{4} \right) \]
Thus,
\[ \lambda_L = \frac{4}{3R} \]

- For the Balmer series (n1=2, n2=3):
\[ \frac{1}{\lambda_B} = R \left( \frac{1}{2^2} - \frac{1}{3^2} \right) = R \left( \frac{1}{4} - \frac{1}{9} \right) = R \left( \frac{9 - 4}{36} \right) = R \left( \frac{5}{36} \right) \]
Thus,
\[ \lambda_B = \frac{36}{5R} \]

### Step 5: Calculate the Frequencies
Frequency \( f \) is related to wavelength \( \lambda \) by the equation:
\[ f = \frac{c}{\lambda} \]
where \( c \) is the speed of light.

- For Lyman series:
\[ f_L = \frac{c}{\lambda_L} = \frac{c}{\frac{4}{3R}} = \frac{3Rc}{4} \]

- For Balmer series:
\[ f_B = \frac{c}{\lambda_B} = \frac{c}{\frac{36}{5R}} = \frac{5Rc}{36} \]

### Step 6: Find the Ratio of Frequencies
Now, we can find the ratio of the frequencies:
\[ \frac{f_B}{f_L} = \frac{\frac{5Rc}{36}}{\frac{3Rc}{4}} = \frac{5}{36} \cdot \frac{4}{3} = \frac{20}{108} = \frac{5}{27} \]

### Conclusion
The ratio of the frequencies of the long wavelength limits of the Balmer and Lyman series of hydrogen is:
\[ \frac{f_B}{f_L} = \frac{5}{27} \]